Modular abelian surfaces of small conductor with nontrivial Tate--Shafarevich groups
We exhibit examples of geometrically simple abelian surfaces $A/\mathbb{Q}$ with conductor bounded by $(10\,000)^2$ whose Tate--Shafarevich groups contain a subgroup isomorphic to $(\mathbb{Z}/p\mathbb{Z})^2$ for each $p = 5, 7, 11, 13$. To find these examples we generalise work of Cremona--Freitas to give a candidate list of all congruences of a certain type between pairs of weight $2$ newforms $f \in S_2^{\mathrm{new}}(\Gamma_0(N))$ and $g \in S_2^{\mathrm{new}}(\Gamma_0(M))$ contained in the LMFDB (i.e., with $N, M \leq 10\,000$) and with coefficient fields of degree $\leq 4$. Passing from the modular forms to the corresponding abelian varieties we use visibility to (unconditionally) prove the existence of non-trivial elements of the Tate--Shafarevich group. Finally we construct an example of an abelian surface with $(\mathbb{Z}/7\mathbb{Z})^2 \subset \mathrm{Sha}(A/\mathbb{Q})$ which is (conjecturally) not visible in any abelian threefold.